   # The Ulam Spiral

### … and many other spirals inspired by it …   ## Step (1)

The table below shows
all the whole numbers squared
in the left column
&
the respective triangular numbers
generated from these
in the right column.

## Step (2)

The tabulation below shows
all the sub-totals of
the triangular numbers from
the right column in
the table in step (1).

## Step (3)

The tabulation below shows
the numbers generated by the
operations described in step (2)
multiplied by 60
& the resulting products
increased by
the respective uneven numbers
starting from 1.

## Step (4)

The tabulation below shows
all the sub-totals of
the numbers per (3),
resulting in all whole numbers
to the 6th. power.

##### Triangularnumber  0

(0 x 1)/2 = 0

1

(1 x2 )/2 = 1

4

(4 x 5)/2 = 10

9

(9 x 10)/2 = 45

16

(16 x 17)/2 = 136

## Step (2)

The tabulation below shows
all the sub-totals of
the triangular numbers from
the right column in
the table in step (1).

0 = 0
0 + 1 = 1
0 + 1 + 10 = 11
0 + 1 + 10 + 45 = 56
0 + 1 + 19 + 45 + 136 = 192

## Step (3)

The tabulation below shows
the numbers generated by the
operations described in step (2)
multiplied by 60
& the resulting products
increased by
the respective uneven numbers
starting from 1.

60(0) + 1 = 1
60(1) + 3 = 63
60(11) + 5 = 665
60(56) + 7 = 3367
60(192) + 9 = 11529

## Step (4)

The tabulation below shows
all the sub-totals of
the numbers per (3),
resulting in all whole numbers
to the 6th. power.

1 = 1 = 16
1 + 63 = 64 = 26
1 + 63 + 665 = 729 = 36
1 + 63 + 665 + 3367 = 4096 = 46
1 + 63 + 665 + 3367 + 11529 = 15625 = 56

From the tabulation above
one can deduce that in general
n to the 6th. power is
the summation of triangular numbers
per the formula shown below: 